Abstract
Optical aspherical lenses with high surface quality are increasingly demanded in several applications in medicine, synchrotron, vision, etc. To reach the requested surface quality, most advanced manufacturing processes are used in closed chain with high precision measurement machines. The measured data are analysed with least squares (LS or L2-norm) or minimum zone (MZ) fitting (also Chebyshev fitting or L∞-norm) algorithms to extract the form error. Performing data fitting according to L∞-norm is more accurate and challenging than L2-norm, since it directly minimizes peak-to-valley (PV). In parallel, reference softgauges are used to assess the performance of the implemented MZ fitting algorithms, according to the F1 algorithm measurement standard, to guarantee their traceability, accuracy and robustness. Reference softgauges usually incorporate multiple parameters related to manufacturing processes, measurement errors, points distribution, etc., to be as close as possible to the real measured data. In this paper, a unique robust approach based on a non-vertex solution is mathematically formulated and implemented for generating reference softgauges for complex shapes. Afterwards, two implemented MZ fitting algorithms (HTR and EPF) were successfully tested on a number of generated reference pairs. The evaluation of their performance was carried out through two metrics: degree of difficulty and performance measure.
Highlights
Laboratoire Commun de Métrologie (LNE-CNAM), 1 Rue Gaston Boissier, 75015 Paris, France; Université Paris-Saclay, Université Sorbonne Paris Nord, ENS Paris-Saclay, LURPA, Department of Mechanical Engineering, College of Engineering, Prince Sattam bin Abdulaziz University (PSAU), Alkharj 16273, Saudi Arabia; Abstract: Optical aspherical lenses with high surface quality are increasingly demanded in several applications in medicine, synchrotron, vision, etc
The generation of reference softgauges dedicated to the assessment of minimum zone (MZ) fitting algorithms of complex shapes is provided
The developed robust approach is based on satisfying KKT first and second order optimality conditions before inferring reference softgauges with non-vertex solutions
Summary
LS or MZ fitting are considered as an optimisation problem. Given a set of N data points, { Pi }1≤i≤ N , let f X, s = 0 is the generic equation describing the shape of the measured surface where Xis the space vector (X = ( x, ŷ, ẑ) in Cartesian coordinates) and s denotes the shape parameters. The aim is to ensure that MZ fitting algorithms return correct ology to assess the results returned by metrology algorithms is to use a reference pair. Reference algorithms do not exist for a wide range of applications in metrology Their development is not always straightforward, especially for applications such as MZ fitting. A common set of data points is submitted to both metrology algorithms (reference algorithm and algorithm under test) and the two results are compared in order to take an accept/reject decision (Figure 3).
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